The Lee "Universal Expander" die is cheap.  But as delivered it only flares the case mouth, without expanding the neck.  How this is useful to cast bullet shooters is beyond me.

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HOWEVER, a little lathe work neatly converts the supplied plug into a proper expander, and expanders for other cases are easily turned up out of bar stock. For some years I've been making custom mandrels for my M dies from a high quality threaded rod, but making custom mandrels for the Lee die is easier.   

Flaring alone is not sufficient.  Sizer dies meant for jacketed bullets leave the neck I.D. 2 or 3 thou smaller than your cast bullet, maybe more if your bullet is sized up to fit the throat.  Flaring alone encourages you to seat your cast bullet into such an undersize neck, but that deforms it; essentially sizing it down just as your lubrisizer did. The bullet is thus undersize when it enters the throat, leaving annular clearance, which will allow it to tip, and/or permit blowby.  A cast bullet should never be tighter than a light drag fit in the neck if you want accuracy.  In revolvers and repeating rifles a crimp is thus necessary to keep it from moving.  Our single shots of course need no crimp.  

The tiny .22 bullet is particularly susceptible to deformation if the neck is too tight.

Lyman mandrels have two diameters. The first diameter opens the whole neck to that drag fit, the second opens it half a thou or so more, to ease the entry of the bullet into the neck. This larger diameter should enter the neck only a few thou. Chamfering the mouth will prevent shaving, unless your seating tools are badly misaligned.

One fault with the Lyman mandrels is that they usually don't come in the optimum size for your bullet and brass.  Preparing .38 Special target ammunition a few years ago I found that I needed different expansion mandrels, depending on brass make and temper, to get the optimum fit.   This is why I started making my own.  Buffalo Bore is offering alternative sizes in the aftermarket, but I don't see any in the smaller sizes.